In Hoffman and Kunze's Linear Algebra, following proof is given for this corollary:
Let A and B be m x n matrices over the field F. Then B is row-equivalent to A if and only if B=PA, where P is a product of m x m elementary matrices.
Proof. (1)Suppose B=PA where P= Es · · · E2E1 and the Ei are m x m elementary matrices. Then E1A is row-equivalent to A, and E2(E1A ) is row-equivalent to E1A . So E2E1A is row-equivalent to A ; and continuing in this way we see that (Es . . . E1)A is row-equivalent to A.
(2)Now suppose that B is row-equivalent to A. Let E1, E2,…,Es
the elementary matrices corresponding to some sequence of elementary row operations which carries A into B. Then B = (Es . . . E1)A
Although I get the corollary intuitively, as each elementary matrix is equivalent to elementary row operation, and thus multiplication of matrix with A is equivalent to series of applications of such elementary row operations. But I don't get how it's being proved here.
- Particularly, how (1) and (2) are connected? Moreover, in (2) we are supposing what we had to actually prove, so how does it help? Also, it sees quite like re-statement of corollary itself.
- In general, what constitutes a mathematical proof? In other words, how do we judge whether it's sufficient and correct? I did go through few questions here, like this one, but how to validate this proof specifically (especially differentiating it with intuition or re-statement)
Best Answer
In an "$X$ if and only if $Y$" proof, you have to prove two separate things: that $X$ implies $Y$, and that $Y$ implies $X$. This is what (1) and (2) are doing, and indeed they are separate proofs. In (1) you assume $X$ and prove $Y$, while in (2) you assume $Y$ and prove $X$.